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Daily Delta-Hedge P&L: Average Approximation and Realized Moves

Article Quant Q&A · Author: Mahi
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Summary

The document examines the difference between an approximate daily P&L formula for a delta-hedged option and P&L calculated from changes in option and stock values. One response explains that the gamma-and-volatility expression gives an average P&L, so it need not match the realized result on each day even when aggregate results are similar.

A second response gives a more path-sensitive, first-order-in-time expression using the squared proportional spot move minus implied variance over the interval, scaled by gamma and spot squared. It identifies this increment with the replication error between the option’s price change and a self-financing delta hedge. The discussion assumes a frictionless hedge with cash transfers between stock and cash accounts; it does not address transaction costs, discrete hedging beyond the daily setting, or higher-order effects.

Key ideas

  • The gamma and volatility expression describes average P&L rather than an exact daily outcome.
  • Realized daily P&L depends on the squared proportional spot move over the interval.
  • The first-order replication error compares option price evolution with a self-financing delta hedge.
  • A self-financing hedge transfers cash between the stock and cash account without outside contributions.

Tags

OptionsVolatilityRisk management
Full text
# Delta hedge value formula


# Delta hedge value formula












When we delta hedge with implied volatility, and dynamically adjust every day, I believe the PnL theoretically is $$PnL = 0.5 \Gamma S^2 (\sigma_r^2 - \sigma_i^2)dt$$ where $\sigma_r$ is realized volatility.

My question is, how accurate is this? I am trying to do a delta hedge experiment, and I find that my daily PnLs range wildly, yet the values given by above formula remain somewhat small (< 1)?

I compute my daily PnL as $$'\text{change in call price'} + \Delta \cdot '\text{change in spot price}'$$ since the first term gives us what we lost/gained through the call, and the second gives us what we earned shorting the stock. But these two formulas don't match .... however the aggregate results do?

## Answer by Mark Joshi (score 4)

https://quant.stackexchange.com/a/33207

$$PnL = 0.5 \Gamma S^2 (\sigma_r^2 - \sigma_i^2)dt $$ this is an average P&L rather than an exact one. So it should agree with your other formula on average but not each day.

## Answer by Quantuple (score 4)

https://quant.stackexchange.com/a/33211

At the order 1 in $dt$ you should rather use $0.5 \Gamma(S, \sigma_i) S^2 ((dS/S)^2 - \sigma_i^2 dt)$ to get the true P&L increment, not the average one (see Mark Joshi's answer).

Also this gives you the replication error i.e. difference between the option price evolution (which you are long) and that of your self-financing delta hedge (= assumes you hedge in a self-financing fashion by transferring cash between stock and cash account with no exogenous infusion/withdrawal of cash).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.